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Electromagnetic Induction Key Concept is Magnetic Flux Faraday’s Law Key Concept is CHANGE in Magnetic Flux A CHANGING ΦB through any closed loop induces” an EMF around the loop. The induced EMF equals the negative of the time rate of change of the total magnetic flux through the loop dΦ ε = − B dt Faraday’s Law ε dΦ = − dt B where Φ B = ∫ surface r r B ⋅ dA = surface It doesn’t matter why the flux changes 1) Constant B, Changing Area: 2) Constant Area, Changing B: 3) Constant Area, Constant B, Changing Cos φ: ∫ BdA cos φ Direction of the Induced EMF’s and Currents In the previous problem, we found the direction of the induced current by noting that the force resulting from the induced current had to oppose the applied force. This observation can be generalized into: Lenz’s Law The direction of any magnetic induction effect is such as to oppose the cause of the effect The “Alternator” (see Y&F example 30-4) The magnetic field r B ,and the Φ B angular frequency ω, are constant = BA By Faraday’s Law: ε = − dΦ dt cos B φ = BA cos = ω AB cos ω t ω t The “Generator” (see Y&F example 30-5) r B ,and the The magnetic field Φ B angular frequency ω, are constant = BA cos φ = BA cos ω t The split-ring commutator “rectifies” the EMF: ε = ω AB cos ω t = ω AB cos ω t Induced Electric Fields Φ B ε = − = BA dΦ B dt = µ = − µ 0 0 nIA nA dI dt The EMF will induce a current that will be indicated by the Galvanometer. This seems very mysterious because there is NO magnetic field outside the solenoid and therefore there can be NO force on the charges inside the conducting loop! Induced Electric Fields Faraday’s Law hold even if there is no Motion and no Magnetic Field Φ B ε = − = BA dΦ B dt = µ = − µ 0 0 nIA nA dI dt Faraday’s Law implies that there is an r r “Induced” Electric Field ∫ E ⋅dl = ε r r dΦ E d l ⋅ = − ∫ dt B This “Induced” Electric Field is a non-electrostatic field that arises, not from static charges, but from a changing B field alone. Maxwell’s Equations The equations below summarize all of the underlying physics of Electricity and Magnetism ∫ r r Q E ⋅ dA = ∫ r r B ⋅ dA = 0 ∫ r r B ⋅ dl = µ ∫ r r dΦ E ⋅ dl = − dt enclosed ε Gauss’s Law 0 Gauss’s Law for magnetism 0 ⎛ ⎜ iC + ε ⎝ 0 dΦ dt E B ⎞ ⎟ ⎠ Ampere’s Law Faraday’s Law AND ( r v v v F = q E + v × B ) Lorentz Force Law Maxwell’s Equations in Free Space If there are no charges and no current’s, Maxwell’s Equations have a very simple and very symmetric form: ∫ r r E ⋅ dA = 0 ∫ r r B ⋅ dA = 0 ∫ r r dΦ B ⋅ = − E d l ∫ dt r r dΦ B ⋅ d l = µ 0ε 0 dt Note that a changing B will induce an E and a changing E will induce a B. This B can in turn induce an E, which will induce a B, and so on… It can be shown that these equations predict the existence of a self-sustaining “wave” that propagates 1 v = with a velocity of: µ ε 0 0 Experimentally this velocity is found to be exactly equal to the speed of light…. All visible light, as well as radio wave, microwaves, x-rays, gamma rays, ultraviolet and infrared radiation are all electromagnetic in origin! E